Operational calculus of linear relations

Abstract

(unbounded spectrum, domain Φ X) in finite-dimens ional spaces which linear operators exhibit only in infinite-dime nsional spaces. We present an outline of the paper. In § 2 we define p(T) where p is a polynomial with coefficients in the field Φ involved in X. We prove that (pq)(T) = p(T)q(T), (poq)(T) = p{q{T)), and point out that sometimes (p + q){T) Φ p(T) + q(T), etc. In § 3 we turn to relations in dual pairs. In this situation, adjoints can be defined. We build an automorphism λ —> λ of Φ into the theory of dual pairs, so as not to exclude the Hubert space situation, which dual pairs are intended to imitate. (Thus the transpose is a special kind of adjoint.) Closedness is defined algebraically, but in a way compatible with the topological concept. Closure of T 7* and other algebraic properties of * are established. Finally, it is shown that if T is closed and its resolvent is not void then p(T) is also closed. Section 4 considers the self-dual case. We give a simple condition (4.3) always true in Hubert space, that T*T be self-adjoint, T being closed. In § 5 we give the spectral analysis of self-ad joint linear relations in Hubert space. In a 1:1 manner these correspond to the unitary operators, via the Cay ley transform. However, it can be shown directly that X is the direct sum of orthogonal subspaces Y, Z which reduce T (= T7*) giving in Za self-ad joint operator and in Fthe inverse of the zero-operator. 2 Linear relations* A relation T between members of a set X and members of a set Y is merely a subset of X x Y. For x e X, T(x) = {y (x, y) e T}. The domain of T consists of those x such that T(x) is not void. T is called single-valued if T(x) never contains more than one element. The range of T is the union of all T(x).